How to deal with random weights initialization in hyperparameters tuning?
I don't think you can.

Say a NN with 3 layers gives an accuracy of 95.3% and another NN with 4 layers gives an accuracy of 95.4%. Then there is no guarantee that the 4 layer NN is better than the 3 layered NN. Since with different initial values the 3 layer NN might perform better.

You could run multiple times and probabilistically say that this is better, but this is computational intense.